### Secularitycondition of the kinetic Carleman system

Pages 33-40

The kinetic theory of gases is considered as a collection of a large number of interacting particles. We consider the discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules, which can have one of two speeds, namely, the Cauchy problem with periodic initial conditions for the system of the Carleman equation. This mathematical model has a number of properties of the Boltzmann equation. This system of equations is a quasi-linear hyperbolic system of partial differential equations. In general, there is no analytic solution for this system. Therefore, under some general assumptions we can find the finite-dimensional approximation of the solutions for the Carleman equation with small Knudsen numbers that allow us to study our problem on the widest scale. Moreover, we can find the secularity condition of the Carleman model. An approximation solution of the Carleman equation for non-periodic initial data will be found in the next article. There is an interesting problem of the existence of the shock waves connecting the pairs of equilibrium states. Here we have a catastrophe theory. It is assumed that the solutions of the Cauchy problem split into the superposition of weakly interacting solitons and decreasing dispersive waves. The Cauchy problem of the Carleman equation is studied for small perturbations of the equilibrium state whereby we have perturbed system. In order to construct the finite-dimensional approximation we use the Fourier method. Construction of finite-dimensional approximation allows doing theoretical studies of solutions for the Cauchy problem of the Carleman equation with small Knudsen numbers.

DOI: 10.22227/1997-0935.2015.7.33-40

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